Modified Kocks–Mecking–Estrin Model to Account Nonlinear Strain Hardening

Abstract

The dislocation density-based model after Kocks–Mecking–Estrin (KME) is widely used to characterize the thermally activated plastic deformation and dislocation kinetics. According to the model, the slope of the stress–strain curve decreases linearly with stress, which contradicts the experimental observation. In the current study, the evolution of dislocation density in the model is generalized to account for the nonlinearity of the slope.

Appendix

The inconsistency in KME model due to fitting procedure is explained below. The model parameters were identified by minimizing the error in predicting the complete stress–strain curve. The parameters thus obtained are used to predict the trend of $\theta$ from Eq. [5]. It was mentioned that all the data points carry equal weightages, and hence the fitting procedure results in a best linear fit of $\theta$ over the entire range. This masks the inaccuracy of KME model at large stress (Figure 2). To further prove this, the experimental data shown in Figure 1 are segmented as shown in Figure A1.

The fitting procedure is repeated for each strain range, and the parameters are tabulated in Table A1.

Fig. A1

Stress–strain data in Fig. 1 segmented to analyze the influence of strain range in parameter identification

Table A1 Dislocation Density Parameters for KME Model Obtained by Fitting Different Strain range

The above parameters are used to plot stress–strain (Figure A2) and $\theta$vs$\sigma$ curves (Figure A3). It is observed that $\theta$ predicted using the parameters for the strain range from 0 to 0.1 correlates with the experimental values of $\theta$ in lower stress range (typically representing the stages II and III hardening). However, the stress–strain curve is accurate only till $\varepsilon =0.1$. On the other hand, parameters obtained over the entire strain range fit the stress–strain curve accurately; however, they average the $\theta$ values. This averaging effect leads to an apparent inconsistency in the general understanding of KME model, as the KME model is known to be accurate at lower stresses and tend to deviate at higher stress range.

Fig. A2

Stress–strain curve predicted using the KME parameters identified by varying the strain range. The predicted curve deviates from the experimental data when extrapolated to higher strain range

Fig. A3

$\theta = \frac{\mathrm{{d}} \sigma }{\mathrm{{d}} \varepsilon }$ is sensitive to the strain range used for parameter identification. $\theta$ predicted is averaged over the data range